We plot the band structure of bearded-edge graphene nanoribbon and winding
number for the bulk Hamiltonian \(H_{\mathrm {bea}}\left( k_{y},k\right) \)  (REF ) as a function of parameter \(k_{y}\)  in Fig. REF (a),
respectively. The winding number we used in this paper is defined as [1]}
\(W=-\frac{i}{4\pi }\int _{BZ}\mathrm {d}k\mathrm {Tr}\left( SQ^{-1}\partial _{k}Q\right) ,\) 
